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Motivated by recent work on optimal approximation by polynomials in the unit disk, we consider the following noncommutative approximation problem: for a polynomial

f

d

freely noncommuting arguments, find a free polynomial

p_{n}

, of degree at most

n

, to minimize

c_{n} ≔ ‖<#comment/> p_{n} f -<#comment/> 1 {‖<#comment/>}^{2}

. (Here the norm is the

{ℓ<#comment/>}^{2}

norm on coefficients.) We show that

c_{n} \to<#comment/> 0

if and only if

f

is nonsingular in a certain nc domain (the row ball), and prove quantitative bounds. As an application, we obtain a new proof of the characterization of polynomials cyclic for the

d

-shift.

Zero-free regions near a line

https://doi.org/10.1007/s00209-023-03364-w

Bickel, Kelly; Pascoe, J E; Sargent, Meredith (November 2023, Mathematische Zeitschrift)

We analyze metrics for how close an entire function of genus one is to having only real roots. These metrics arise from truncated Hankel matrix positivity-type conditions built from power series coefficients at each real point. Specifically, if such a function satisfies our positivity conditions and has well-spaced zeros, we show that all of its zeros have to (in some explicitly quantified sense) be far away from the real axis. The obvious interesting example arises from the Riemann zeta function, where our positivity conditions yield a family of relaxations of the Riemann hypothesis. One might guess that as we tighten our relaxation, the zeros of the zeta function must be close to the critical line. We show that the opposite occurs: any poten- tial non-real zeros are forced to be farther and farther away from the critical line.

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